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F.H.R. 1 Trigonometric functions The general expression for angles with a given trigonometric ratio: The topic of this lesson is functions in which the trigonometric ratios of the unknown quantity occur. Equations of this type may be reduced to the solution of one or more equations of the type: Sin x = , Cos x = or Tan x = In this kind of equation, is known and x is the unknown element. Example 1. Consider the equation Tan x = 1 One solution is Tan x = 45 but that is not the only solution because it is a fact that: (Tan = 45 + 180 ) = Tan 45 = 1 = 225 is another solution to this equation. The general solution is: = 45 + n 180 where n is an integer, either negative, 0 or positive. By giving n different values, different solutions to the equation are obtained. Figure 1. Graph of y = Tan x F.H.R. 2 Reference to figure 1 may help showing this. The horizontal line y = is drawn on the same scale and intersects the tangent curve in the points A, A’, A”. B, B’ etc. If the absissa of A = , that of A’ is + 180, of A” = + 2 x 180 For the equation Cos = it must be realized that if is numerically greater than 1, the equation will have no solution. When is numerically less than 1, follow this procedure: figure 2 shows the graph of y = cos and the line y = is drawn on the same scale. The abscissae of the point of intersection will give the solution to the equation Cos = Figure 2. Graph of y = cos x If is the smallest angle for which y = cos the absicca of A, A’, A” etc are 360 - , 360 + , etc. These are all particular cases of the formula: x = n360 - , In this equation n is an integer, positive, 0 or negative. The procedure for Sin x = is the same as cos x = The general solution to the equation Sin x = is x = n180 + (-1)n , n is an integer, positive, 0 or negative. F.H.R. Figure 3. 3 Graph of y = sin x Example 2 Find the general solution to the equation Sin x = 0.515 which lay in the range 0 to 360 state the general solution Solution: From your calculators you will find that Sin 31 = 0.515, therefore 31 is a solution. The general solution is however: x = n 180 + (-1)n 31. With n = 0 or 1 the solution in the range 0 to 360 is obtained. That is 31 and 180 - 31 = 149